$(10$ pt each$)$ Consider a three$-$month forward contract on gold. Suppose that it costs $$3$ per ounce per quarter to store gold with payment being made at the beginning of each quarter. Assume that the spot price is $$400$ per ounce and the risk$-$free rate is $6\%(4).$

$($a$)$ What is the theoretical forward price that avoids artibrage profit opportunity??

$($b$)$ What are the possible arbitrage opportunities when the forward price is $$420?$